K-theory, Dynamics, and Intersection

PI提出了两个项目，接口代数和微分拓扑和K理论。其中第一个目的是加强扭转不变量和动力系统中闭合轨道的计数之间的关系。这里的目标是推广将Reidemeister挠率与Lefschetz zeta函数联系起来的Milnor方程。该项目的一个方面将导致这个方程的一个版本，它适用于动力系统的家庭，将涉及一个更高的K-理论不变的计数周期轨道。主要的工具将来自瓦尔德豪森的工作代数K理论和函子TR，这是拓扑学家的版本的拓扑德拉姆-维特复形。第二个项目将研究交叉理论的多相对版本，这将提供超越亚稳态范围的见解。我们将研究将映射从一个流形变形到另一个流形的障碍，假设映射可以从任何真子流形的并集变形。障碍物将存在于边界群的直接和中。在一定的范围内，障碍物将捕捉整个故事。这一理论的应用将在嵌入理论和链接同伦理论中给出。所提出的研究主要是关于“流形”，这是满足一定的齐性性质的拓扑空间。局部地说，所有的流形都是相似的，因为在任何一点上都可以看到欧几里得空间的副本。正是流形的整体结构使它们成为有趣的研究对象。通常，代数拓扑学家通过分配某些代数量（称为“不变量”）来研究流形，这些代数量测量全局拓扑结构。具有不同不变量的流形可以彼此区分。流形在物理学、化学和生物学中自然出现，作为一组适当的“好”代数方程的解的空间，模拟科学研究对象（时空、原子、动力系统等）。.流形在数学中起着核心作用。通常情况下，关于流形的数学问题可以用与流形相关的空间之间的参数化函数族来表述。同伦理论是一门旨在解决这类函数族问题的学科。K-理论是一种代数理论，它是流形不变量的容器。PI将研究可以使用同伦理论和K理论分析的某些类型的流形问题。

The PI proposes two projects that interface algebraic and differential topology and K-theory. The first of these aims to strengthen the relationship between torsion invariants and the counting of closed orbits in dynamical systems. The goal here is to generalize Milnor's equation relating Reidemeister torsion to the Lefschetz zeta function. One aspect of the project will lead to a version of this equation which holds for families of dynamical systems that will relate a higher K-theory invariant to an invariant counting periodic orbits. The main tools will come from Waldhausen's work on algebraic K-theory and the functor TR, which is the topologists' version of the topological de Rham-Witt complex. The second project will study multi-relative version of intersection theory which will give insights beyond the metastable range. We will study obstructions to deforming a map from a manifold into another one off of a finite collection of pair-wise disjoint submanifolds, assuming that the map can be deformed off of the union of any proper sub-collection. The obstruction will live in a direct sum of bordism groups. In a certain range, the obstruction will capture the entire story. Applications of this theory will be given in embedding theory and also in the theory of link homotopy.The proposed research is largely about "manifolds" which are topological spaces that satisfy a certain homogeneity property. Locally speaking, all manifolds are alike in that at any point one sees a copy of Euclidean space. It is the global structure of manifolds that makes them interesting objects of study. Typically, algebraic topologists study manifolds by assigning certain algebraic quantities, called "invariants," to them, which measure global topological structure. Manifolds having different invariants can then be distinguished from one another. Manifolds arise naturally in physics, chemistry and biology as spaces of solutions of a suitably "nice" set of algebraic equations modeling the scientific object of study (space-time, atoms, dynamical systems, etc.) . Manifolds play a central role in mathematics. It is often the case that mathematical questions about manifolds can be formulated in terms of parametrized families of functions between spaces associated with manifolds. Homotopy theory is a subject designed to tackle questions about such families of functions. K-theory is an algebraic theory which is a receptacle for invariants of manifolds. The PI will research certain kinds of manifold questions which can be analyzed using homotopy theory and K-theory.